Dealing with periodic boundary conditions for 1D, 2D and 3D isogeometric finite element method

Marcin Łoś¹ , Maciej Paszyński¹, Lisandro Dalcin², Victor Calo²

¹AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland.
²King Abdullah University of Science and Technology, Thuwal, Saudi Arabia.

DOI:

https://doi.org/10.7494/cmms.2015.1.0525

Abstract:

In this paper we analyze the problem of implementing periodic boundary conditions in the isogeometric finite element method (ISO-FEM). The ISO-FEM method uses the B-spline-based basis functions, which facilitates usage of the same basis functions for approximation of the geometry as well as for the numerical solution of the modeled physical phenomena. The usage of the B-spline based basis functions results in C^(p-1) global continuity of the solution. The drawback is a difficulty in implementing the periodic boundary conditions, and special dedicated methods are necessary. In this paper we present two algorithms implementing the periodic boundary conditions. The first one is an iterative algorithm that utilizes widely available block-diagonal LAPACK solver. The second one is a modification of the multi-frontal solver algorithm itself, and it requires a dedicated solver with its source code modified accordingly. The presented methods can be applied in one, two or three-dimensional isogeometric finite element method.

Cite as:

Łoś, M., Paszyński, M., Dalcin, L., & Calo, V. (2015). Dealing with periodic boundary conditions for 1D, 2D and 3D isogeometric finite element method. Computer Methods in Materials Science, 15(1), 213-218. https://doi.org/10.7494/cmms.2015.1.0525

Article (PDF):

Keywords:

Isogeomtric analysis, Finite element method, Periodic boundary conditions, Direct solvers

Publication dates:

Received: 16.09.2014, accepted: 10.12.2014, published: 10.01.2015

Publication type:

Original scientific paper

References:

Collier, N., Pardo, D., Dalcin, L., Paszynski, M.,Calo, V.M., 2012, The Cost of Continuity: A Study of the Performance of Isogeometric Finite Elements Using Direct Solvers, Comput Method Appl M, 213-216, 353-361.

Cottrel, J.A., Hughes, T.J.R., Bazilevs, Y., 2009, Isogeometric Analysis. Toward Integration of CAD and FEA, John Willey & Sons, Chichester, United Kingdom.

LAPACK – Linear Algebra PACKage 3.5.0, http://www.netlib.org/lapack, November 2014.

Obrok, P., Pierzchała, P., Szymczak, A., Paszyński, M., 2010, Graph Grammar-Based Multi-Thread Multi-Frontal Parallel Solver with Trace Theory-Based Scheduler, Procedia Computer Science, 1, 1993-2001.

Wozniak, M., Kuznik, K., Paszynski, M., Calo, V. M., Pardo, D., 2014, Computational Cost Estimates for Parallel Shared Memory Isogeometric Multi-Frontal Solvers, Comput Math Appl, 67, 1864-1883.